Several adaptive techniques exist for local truncation error estimations, e.g. Taylor series and Richardson extrapolation as well as RK [1]}. Runge–Kutta formulas can be used to control error and step-size at every step and make step rejection less expensive.
This development considers adaptive technique based on Runge–Kutta (RK) methods where two RK methods, called embedded Runge–Kutta pairs, one more accurate than the other, are appropriately chosen for which the higher-order method is assumed to approximate (or be very close to) to the unknown exact solution.
With this assumption, the difference between the two numerical solutions gives a measurement for the local error. This local truncation error estimate is used to adjust the step-size.
An additional advantage of embedded Runge–Kutta methods, compared with Taylor series and Richardson extrapolation, is that the re-evaluation of the stage solutions is no longer necessary. The pairs share the same stage computations, i.e., they have the same \(A\) . This essentially provides the local error estimation at little to no cost.
